Properties

Label 6.396259.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 396259 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$396259 $
Artin number field: Splitting field of $f= x^{7} - x^{4} - x^{3} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.396259.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: $ x^{2} + 102 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 7 + 25\cdot 103 + 98\cdot 103^{2} + 36\cdot 103^{3} + 98\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 78 a + 69 + \left(66 a + 68\right)\cdot 103 + \left(48 a + 90\right)\cdot 103^{2} + \left(44 a + 93\right)\cdot 103^{3} + \left(16 a + 33\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 4 a + 81 + \left(57 a + 5\right)\cdot 103 + \left(43 a + 70\right)\cdot 103^{2} + \left(74 a + 19\right)\cdot 103^{3} + \left(6 a + 80\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 21 a + 1 + \left(30 a + 42\right)\cdot 103 + \left(96 a + 30\right)\cdot 103^{2} + 5\cdot 103^{3} + \left(53 a + 13\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 25 a + 44 + \left(36 a + 57\right)\cdot 103 + \left(54 a + 72\right)\cdot 103^{2} + \left(58 a + 89\right)\cdot 103^{3} + \left(86 a + 5\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 82 a + 22 + \left(72 a + 51\right)\cdot 103 + \left(6 a + 96\right)\cdot 103^{2} + \left(102 a + 12\right)\cdot 103^{3} + \left(49 a + 65\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 99 a + 85 + \left(45 a + 58\right)\cdot 103 + \left(59 a + 56\right)\cdot 103^{2} + \left(28 a + 50\right)\cdot 103^{3} + \left(96 a + 12\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

Cycle notation
$(1,2,3,4,5,6,7)$
$(1,2)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character value
$1$$1$$()$$6$
$21$$2$$(1,2)$$4$
$105$$2$$(1,2)(3,4)(5,6)$$0$
$105$$2$$(1,2)(3,4)$$2$
$70$$3$$(1,2,3)$$3$
$280$$3$$(1,2,3)(4,5,6)$$0$
$210$$4$$(1,2,3,4)$$2$
$630$$4$$(1,2,3,4)(5,6)$$0$
$504$$5$$(1,2,3,4,5)$$1$
$210$$6$$(1,2,3)(4,5)(6,7)$$-1$
$420$$6$$(1,2,3)(4,5)$$1$
$840$$6$$(1,2,3,4,5,6)$$0$
$720$$7$$(1,2,3,4,5,6,7)$$-1$
$504$$10$$(1,2,3,4,5)(6,7)$$-1$
$420$$12$$(1,2,3,4)(5,6,7)$$-1$
The blue line marks the conjugacy class containing complex conjugation.